Week 4 Statistics 2 Questions

Q1

Let

X

be a continuous random variable with the following CDF:

F_X(x)=\begin{cases}0, & x<-1\\\frac{(x+1)^2}{2}, & -1\le x<0\\1-\frac{(1-x)^2}{2}, & 0\le x<1\\1, & x\ge 1\end{cases}

Which of the following is the correct PDF of

X

?

Single correctOCTOBER 26, 2025

OCTOBER 26, 2025

Q2

Find the value of

k

.

ComprehensionOCTOBER 26, 2025

OCTOBER 26, 2025

Q3

Find the value of

P(1<X<2 \mid X>\frac{1}{2})

. Enter the answer correct to two decimal places.

ComprehensionOCTOBER 26, 2025

OCTOBER 26, 2025

Q4

The time (in minutes) a customer spends inside a service center follows an exponential distribution with an average of

10

minutes. What is the probability that a customer has to wait at least

12

more minutes in the center given that he has already spent

10

minutes in the center? Enter the answer correct to two decimal places.

Integer answerJULY 13, 2025

JULY 13, 2025

Q5

Calculate the value of

\operatorname{Var}(X)

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q6

Define

Y=72X+5

, then find the value of

\operatorname{Var}(Y)

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q7

Find the value of

a+b

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q8

If the value of

b

is

14

, then find the value of

P(5<T<11\mid T>7)

. Enter the answer correct to two decimal places.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q9

The probability density function of a random variable

X

is given as

f_X(x)=\begin{cases}c, & 0\le x<\frac{1}{4}\\2c, & \frac{1}{4}\le x<\frac{3}{4}\\3c, & \frac{3}{4}\le x<1\\0, & \text{otherwise}\end{cases}

Find the value of

c

. Enter the answer correct to one decimal place.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q10

Find the value of

a

.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q11

What is the CDF of

X

?

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q12

What is the value of

P(-4<X\le6)

?

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q13

Find the value of

k

. Enter the answer correct to two decimal places.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q14

Calculate the probability that the length of a randomly selected rod is between

2

cm and

4

cm. Enter the answer correct to two decimal places.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q15

A tech company develops a new software system. The time until the system experiences its first major failure is modeled by an exponential distribution with an average failure time of

5

years. Find the value of

k

such that there is a

95\%

chance the system will fail within

k

years.

Single correctOCTOBER 27, 2024

OCTOBER 27, 2024

Q16

Suppose that the number of miles that a car can run before its battery wears out is exponentially distributed, with an average value of

10000

miles. If you desire to take a

5000

-mile trip, what is the probability that you will be able to complete the trip without replacing the car battery?

Single correctJULY 07, 2024

JULY 07, 2024

Q17

A teacher observes that the cumulative distribution function (CDF) for the scores on a mathematics test is

F(x)=\begin{cases}0, & x<0,\\ x^2, & 0\le x\le1,\\ 1, & x>1.\end{cases}

For

0\le x\le1

, where

x

is the score normalized to

1

. If a student scores above

0.7

, what is the conditional probability that they actually score above

0.9

? Enter the answer correct to two decimal places.

Integer answerJULY 07, 2024

JULY 07, 2024

Q18

Find the value of

c

so that

f_X(x)

is a valid PDF.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q19

Calculate the CDF of

X

.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q20

The CDF of a random variable

X

is given as

F_X(x)=1-e^{-4x}

for

x\ge0

, and

0

otherwise. What is the value of

P(-4<X\le6)

?

Single correctFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q21

Find the value of

k

. Enter the answer correct to one decimal place.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q22

What is the value of

P(1<X<2.5)

? Enter the answer correct to two decimal places.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q23

Consider a function

f:\mathbb{R}\to\mathbb{R}

such that

f(x)=cx^2

for

1\le x<2

,

f(x)=cx

for

2\le x<4

, and

f(x)=0

otherwise, where

c

is any real constant. Find

c

such that

f

is a valid density function. Enter the answer correct to two decimal places. Hint:

\int x^k\,dx=\frac{x^{k+1}}{k+1}

.

Integer answerOCTOBER 29, 2023

OCTOBER 29, 2023

Q24

Which of the following is true?

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q25

Find

P(|X|<\frac{\alpha}{2})

. Enter the answer correct to two decimal places.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q26

Let

X

be a continuous random variable with the following PDF:

f_X(x)=\frac{k}{(1+x)^2}

for

0\le x\le 4

, and

0

otherwise. Find the value of

k

. Enter the answer correct to two decimal places. Hint:

\int\frac{1}{(a+bx)^2}\,dx=-\frac{1}{b(a+bx)}

.

Integer answerJULY 16, 2023

JULY 16, 2023

Q27

Find the value of

k

.

ComprehensionJULY 16, 2023

JULY 16, 2023

Q28

Find the value of

P(\frac{1}{4}\le X\le \frac{3}{4})

. Enter the answer correct to two decimal places.

ComprehensionJULY 16, 2023

JULY 16, 2023

Q29

Which of the following statements are correct?

Single correctOCTOBER 16, 2022

OCTOBER 16, 2022

Q30

Among the options below, for what values of

a

and

b

is the function

f

a valid density function?

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q31

With the choice of

a

and

b

given in the previous question, find

P\left(X\le -\frac{1}{2}\mid X<1\right)

. Enter the answer correct to three decimal places.

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q32

Let

f:\mathbb{R}\to\mathbb{R}

be defined by

f(x)=\begin{cases}0, & -\infty < x < 0,\\ \frac{1}{4}, & 0 < x \le 1,\\ \frac{3}{4x^2}, & 1 < x < \infty.\end{cases}

If a random variable

X

has the PDF

f(x)

, calculate

P\left(-\frac{1}{2}<X<\frac{3}{2}\right)

. Enter the answer correct to one decimal place. Hint: use

\int_a^b \frac{1}{x^2}\,dx = \frac{1}{a}-\frac{1}{b}

.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q33

Find the value of

p

so that

f_X(x)

is a valid PDF. Enter the answer correct to two decimal places.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q34

Find the value of

P(X<4\mid X>2)

. Enter the answer correct to two decimal places.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q35

What is the probability that Jadeja will bat for more than

40

minutes? Enter the answer correct to two decimal places.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q36

What is the probability that Jadeja will bat for more than

60

minutes given that he has already batted for more than

40

minutes? Enter the answer correct to two decimal places.

Integer answerJUNE 05, 2022

JUNE 05, 2022

All questions